A357318 Decimal expansion of 1/(2*L), where L is the conjectured Landau's constant A081760.
9, 2, 0, 3, 7, 1, 3, 7, 3, 3, 1, 7, 9, 4, 2, 4, 9, 7, 6, 5, 5, 5, 1, 8, 5, 6, 4, 5, 4, 3, 1, 7, 2, 9, 9, 4, 7, 2, 6, 2, 4, 5, 7, 9, 1, 9, 4, 9, 8, 9, 4, 3, 3, 8, 3, 4, 3, 3, 0, 0, 1, 9, 9, 7, 7, 3, 1, 0, 1, 8, 0, 8, 0, 8, 0, 5, 6, 8, 5, 6, 3, 9, 3, 6, 3, 3, 8, 5
Offset: 0
Examples
0.9203713733179424976555185645431729947262...
Links
- Markus Faulhuber, An application of hypergeometric functions to heat kernels on rectangular and hexagonal tori and a "Weltkonstante"-or-how Ramanujan split temperatures, The Ramanujan Journal volume 54, pages 1-27 (2021). See p. 23.
- Markus Faulhuber, Anupam Gumber, and Irina Shafkulovska, The AGM of Gauss, Ramanujan's corresponding theory, and spectral bounds of self-adjoint operators, arXiv:2209.04202 [math.CA], 2022, p. 2.
Programs
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Mathematica
First[RealDigits[N[Gamma[1/6]/(2Gamma[1/3]Gamma[5/6]),88]]]
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PARI
1/(2*gamma(1/3)*gamma(5/6)/gamma(1/6)) \\ Michel Marcus, Sep 24 2022
Formula
Equals Sum_{k,m in Z^2} exp(-Pi*(2/sqrt(3))*(k^2+k*m+m^2))*exp(2*Pi*i*(k/3-m/3)).
Equals Sum_{k>=0} (binomial(-1/3,2*k)^2 - binomial(-1/3,2*k+1)^2). - Gerry Martens, Jul 24 2023
Equals 3*Gamma(1/3)^3 / (2^(8/3) * Pi^2). - Vaclav Kotesovec, Jul 27 2023